Reverse isoperimetric problems under curvature constraints

Abstract

In this paper we solve several reverse isoperimetric problems in the class of λ-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some λ > 0.We give an affirmative answer in R 3 to a conjecture due to Borisenko which states that the λ-convex lens, i.e., the intersection of two balls of radius 1/λ, is the unique minimizer of volume among all λ-convex bodies of given surface area.Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the λ-convex lens (properly defined in non-zero curvature spaces) has the smallest inscribed ball among all λ-convex bodies of given surface area. This solves a conjecture due to Bezdek on minimal inradius of isoperimetric ball-polyhedra in R n .